Embed Size (px)
Citation preview
CHROMATIC DISPERSION
COMPENSATION BY SIGNAL
PREDISTORTION:
LINEAR AND NONLINEAR
FILTERING
Shaham Sharifian
Communications Systems GroupDepartment of Signals and SystemsCHALMERS UNIVERSITY OF TECHNOLOGY
Goteborg, Sweden EX026/2010
Abstract
Chromatic dispersion is one of the most significant impairments in long-distancefiber-optical communications. Electrical domain compensation of optical chromaticdispersion by signal predistortion using digital processing is investigated with twodifferent approaches: using a linear digital FIR compensating filter and using anonlinear filter based on a look-up table (LUT). The linear compensating system issimulated with quadrature phase shift keying (QPSK) and 16-quadrature amplitudemodulation (16-QAM) schemes. In the 16-QAM system the maximum amplitudeof the signal generated after the compensating filter can be large, so the maximumamplitude is calculated and is employed in compensating the cosine function of theCartesian Mach-Zehnder modulator. The time and frequency domain implementa-tion of the linear compensating filter are considered more carefully and the numberof adjacent symbols which contribute to the intersymbol interference is calculated.
In [1] it has been shown that the LUT generated for compensating chromatic dis-persion can be significantly compressed with the Hadamard transform. Neverthe-less there are still some constraints in implementing such a system. These con-straints, which are mainly memory constraints and processing time constraints, areintroduced and some implementation details are discussed to avoid the memoryconstraints, e.g., the need for generating the large-size LUT before generating thecompressed LUT. Removing this constraint allows us to go for longer input symbolsequences to the LUT. For example at the distance of 500 km in a 10 Gbaud/secQPSK system, the chromatic dispersion can be completely compensated by tak-ing 11 symbols as the input to the LUT. It is also shown that using at least a 4-bitdigital-to-analog converter (DAC) in a QPSK system and a 5-bit DAC in a 16-QAMsystem does not lead to degradation of their performances.
The linear compensating filter can compensate any amount of dispersion in bothmodulation schemes with a reasonable number of filter taps. The LUT-based com-pensating system, with or without the Hadamard transform, has the same perfor-mance as the linear system, i.e., it can compensate CD completely, if the length ofthe input symbol stream to the LUT is chosen at least equal to the number of ad-jacent symbols which contribute to the intersymbol interference. For shorter inputsequences to the LUT, we see degradation in the performance of the system, whichcan now be improved by the Hadamard transform, as previously shown in [1].
Keywords: Chromatic dispersion (CD), predistortion, electronic linear precom-pensation, nonlinear precompensation, look-up table, Hadamard transform, opticalfiber communication, optical modulation formats, coherent detection, digital signalprocessing (DSP) .
iii
iv
Acknowledgments
Hereby I would like to thank my supervisor Prof. Erik Agrell for his support andinsightful guidance. I also want to express my gratitude to the Ph.D. student ofcommunication system group, Lotfollah Beygi, for always being available for varioususeful discussions. Thanks also to Dr. Pontus Johannisson and Prof. MagnusKarlsson at MC2 department for discussions about chromatic dispersion.
Goteborg, August 20, 2010
SHAHAM SHARIFIAN
NOTATION
AWGN additive white Gaussian noiseCD chromatic dispersionDCF dispersion compensating fibersDDMZM dual drive Mach-Zehnder modulatorDSP digital signal processingEDC electronic dispersion compensationEDFA Erbium doped fiber amplifierEPD electronic predistortionGVD group velocity dispersionISI intersymbol interferenceLUT look-up tableMMF multimode fibersOOK on-off keyingPMD polarization mode dispersionQAM quadrature amplitude modulationRRC root raised-cosineSMF single-mode fibersSNR signal-to-noise ratio
v
Contents
1 INTRODUCTION 1
2 FIBER-OPTIC COMMUNICATION SYSTEM 4
2.1 OPTICAL TRANSMITTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 THE MODULATOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 MODULATION SCHEMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 PULSE-SHAPING FILTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 THE OPTICAL CHANNEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 CHROMATIC DISPERSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 TIME AND FREQUENCY RESPONSE . . . . . . . . . . . . . . . . . . . . . 11
2.3 THE OPTICAL COHERENT RECEIVER AND DETECTION. . . . . . . . . . 11
2.4 LITERATURE STUDY ON CHROMATIC DISPERSION COMPENSATION 17
3 CHROMATIC DISPERSION COMPENSATION BY LINEAR FILTERING 19
3.1 SYSTEM MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 NUMBER OF FILTER TAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 TIME-DOMAIN AND FREQUENCY-DOMAIN IMPLEMENTATIONS . . 21
3.4 EFFECT OF FILTERING ON THE AMPLITUDE . . . . . . . . . . . . . . . . . . . 23
3.5 PERFORMANCE OF LINEAR PRECOMPENSATING METHOD . . . . . . 24
3.5.1 THE EFFECT OF D/A RESOLUTION ON THE PERFORMANCE 25
4 CHROMATIC DISPERSION COMPENSATION BY NONLINEAR FILTERING 28
4.1 THE TRANSMITTER SYSTEM MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 THE CONSTRAINTS AND THE SOLUTIONS . . . . . . . . . . . . . . . . . . . . . 28
4.3 PERFORMANCE OF NONLINEAR PRECOMPENSATING METHOD . . 32
4.4 COMPARING WITH LINEAR COMPENSATION, PROS AND CONS . . 33
vi
CONTENTS vii
5 CONCLUSIONS 35
1 INTRODUCTION
A communication system is for transmitting information from one place to anotherby modulating an electromagnetic carrier wave. The carrier wave frequency variesfrom one system to another. Optical communication systems use high carrier fre-quencies (∼100 THz) in the visible or ultraviolet region of the electromagnetic spec-trum while wireless communication systems and microwave systems use lower car-rier frequencies (∼1-10 GHz). Optical communication systems are sometimes calledlightwave systems and when they employ optical fibers as the medium for transmit-ting information, they are called fiber-optic communication systems.
In a fiber-optic communication system different modulation formats can be usedfor modulating and sending pulses of light through an optical fiber. On-off keying(OOK) modulation was mostly used in digital lightwave systems until a few yearsago because of its simplicity. As binary modulation formats have a low spectral effi-ciency, some advanced modulation schemes with higher spectral efficiency are moreenticing to use. These multilevel modulations need coherent detection. Althoughcoherent receivers increase complexity, they allow more advanced modulation tech-niques which leads to higher spectral efficiencies.
Compared to the bandwidth of radio communication systems (which is in rangeof ∼MHz), much higher bandwidth is available in fiber-optic systems (∼THz) dueto the much higher carrier frequency in fiber-optic systems than in wireless radiosystems. This results in higher transmission data rates in fiber-optic communicationsystems (∼Tb/s) compared to wireless radio communication systems (∼Mb/s).
Supporting high data rates in long-distance transmissions, fiber-optic communica-tion systems have revolutionized the telecommunication industry and they have beenwidely used in long-distance communications such as intercontinental, transatlanticand transpacific communications. Today most of the copper wires in core of globaltelecommunication infrastructure are replaced by optical fibers due to much lowerattenuation and interference.
An optical fiber consists of three parts: a core, a cladding, and a buffer (an outercoating to protect cladding). The cladding and the core are usually made from highquality silica glass or sometimes from plastic. The cladding has a lower refractiveindex, so the light is guided along the core by the total internal refraction method.Multimode fibers (MMF) and single-mode fibers (SMF) are two main types of op-tical fibers used in fiber-optic communications. SMF has a smaller core radius (<10 micrometers) than MMF (≥ 50 micrometers). SMF interconnectors and compo-nents are usually more expensive but the performance of this type of link is higherthan MMF link. MMF links have higher attenuation and because of introducingmultimode distortion, they often limit the bandwidth and length of the link. Inthis thesis we consider SMF as the transmission medium. In such a medium thefollowing impairments limit the performance of the system [2, pp. 39–64]:
1
2 Chapter 1 Introduction
• Fiber losses limit the transmission distance.
• Dispersion limits the bit rate due to pulse broadening.
• Nonlinear effects limit the system performance due to distortion of the signal.
Fiber losses are because of material absorption, scattering, or waveguide imperfec-tions and they attenuate the average power reaching the receiver. To compensatethis attenuation optical amplifiers are used periodically. In long-haul transmissionsystems, the spacing between repeaters is mainly determined by the amount of fiberloss. The bandwidth that can be transmitted through the optical-fiber is consider-ably limited by chromatic dispersion. There are two kinds of dispersion: chromaticdispersion (CD) and polarization mode dispersion (PMD). The refractive index ofthe fiber is frequency dependent, so different frequency components travel at differ-ent speeds and they arrive at different times at the output end which leads to pulsebroadening and intersymbol interference (ISI). This phenomenon is known as chro-matic dispersion. The components that are orthogonally polarized also travel at dif-ferent speeds due to manufacturing defects, stress, vibration or other imperfectionswhich causes polarization mode dispersion. PMD can degrade system performanceconsiderable especially for old fibers and at high data rates. Major nonlinear effectsin optical fibers are stimulated Raman scattering, stimulated Brillouin scattering,self-phase modulation (SPM), cross-phase modulation, and four-wave mixing.
In this thesis we do not consider nonlinear effects and we just deal with chromaticdispersion which is a linear phenomenon. Chromatic dispersion limits the bandwidthand thus the transmission data rate because of pulse spreading. The chromatic dis-persion can be compensated in both electrical and optical domains. The main goalof this thesis is designing linear and nonlinear electronic dispersion compensatorsystems using a finite impulse response (FIR) filter and a look-up table (LUT) re-spectively and analyzing and comparing the two systems. Some implementationconsideration on the linear system is also discussed. Chapter 1 introduces the fiber-optic communication system, the impairments that exist in such a system and themain goal of the thesis. The fiber-optic communication system model is describedin Chapter 2, where the optical transmitter, channel, and receiver are reviewed. To-ward the end of this chapter different modulation formats and pulse-shaping filtersare introduced and dispersion is studied more carefully. The chapter ends with aliterature study on different techniques that has been used to compensate fiber im-pairments, particularly fiber dispersion. The linear system design for compensatingCD is explained in Chapter 3. In this chapter, we consider a real-time linear finiteimpulse response (FIR) filter to compensate CD. This filter can be implementedin both time and frequency domain. A lower bound on the number of filter tapsrequired for compensating chromatic dispersion as a function of sampling frequencyand pulse-shaping filter bandwidth is derived. In addition, an expression for thenumber of adjacent symbols which contribute to the ISI is given and the maximumamplitude generated after the FIR compensator filter is calculated and used in com-pensating the modulator effect. The performance of the system is shown and the
3
effect of digital-to-analog resolution on the performance is investigated. The nonlin-ear filtering design is discussed in Chapter 4. In [1] it has been shown that the LUTgenerated for compensating chromatic dispersion can be significantly compressedwith the Hadamard transform. Nevertheless there are still some constraints, suchas memory constraints, in implementing such a system. In this chapter some imple-mentation details are discussed to avoid the memory constraints such as the needfor generating the large-size LUT before generating the compressed LUT. Moreover,at the end of this chapter the linear and nonlinear implemented systems are com-pared in sense of performance, memory requirement and computational complexity.Finally this thesis work is concluded with some conclusions in Chapter 5.
2 FIBER-OPTIC COMMUNICATION SYS-TEM
2.1 OPTICAL TRANSMITTER
Common optical sources in fiber-optic communication systems are semiconductoroptical sources such as light-emitting diodes (LEDs) or semiconductor lasers whichare also known as laser diodes or injection lasers. The electrical input signal mustbe converted to an optical signal before launching it into the optical fiber. Thisis done by modulating the optical source and can be accomplished in two ways:direct modulation or using an external optical modulator. In direct modulation, thelight source’s driving current is directly modulated by the electrical drive signals.Although this method is less complex, it is not achievable in high-speed transmissionsdue to its large spectral width. So in applications with high bit rates, an externalmodulator is applied to modulate the light carrier by the electrical data signal.
2.1.1 THE MODULATOR
An external modulator that has been widely used is the dual drive Mach-Zehndermodulator (DDMZM). As shown in Figure 2.1, the DDMZM divides the incom-ing optical signal into two signals. Each wave propagates through one arm of theDDMZM. Applying voltage to each arm results in a phase shift in the optical signal.Then the phase shifted waves are added to form the modulated optical signal.
The transfer function of DDMZM is [3]:
Eout(t) =1
2
(
Ein(t)ejπ
V1(t)Vπ
)
+1
2
(
Ein(t)ejπ
V2(t)Vπ
)
(2.1)
= Ein(t) cos
(
π
2· V1(t)− V2(t)
Vπ
)
ejπV1(t)+V2(t)
2Vπ ,
where Ein is the input optical signal to the modulator, Eout is the output optical
LASER + LASEREinEin
Ein/2
Ein/2
EoutEout
V1(t)
V1(t)
V2(t)
V2(t)
ejπV1(t)Vπ
ejπV2(t)Vπ
Figure 2.1: Block diagram of dual drive Mach-Zehnder [1].
2.1 OPTICAL TRANSMITTER 5
signal from the modulator, V1(t) and V2(t) are the drive signals to the two arms ofDDMZM and Vπ is a reference voltage. Vπ is typically around 4 volts.
It can be seen from above equation that DDMZM can modulate both amplitudeand phase of the optical signal. If the same drive signal is applied to both arms,i.e., when V1(t) = V2(t), the DDMZM operates as a phase modulator (PM) whilewhen V1(t) = −V2(t) the DDMZM acts as an amplitude modulator (AM) and thetransfer function is simplified to
Eout(t) = Ein(t) cos
(
πV1(t)
Vπ
)
. (2.2)
2.1.2 MODULATION SCHEMES
The optical carrier before modulation has the form [2, p. 14] :
Ein(t) = eAc cos(ωct+ ϕ), (2.3)
where Ein(t) is the electric field vector, e is the polarization unit vector (if e is ascalar unit it results in the scalar Ein(t)), Ac is the amplitude, ωc is the carrierfrequency and ϕ is the carrier phase. So different modulation formats can be usedas follows:
• Amplitude-shift keying (ASK), in which Ac is modulated
• Frequency-shift keying (FSK), in which ωc is modulated
• Phase-shift keying (PSK), in which ϕ is modulated
• Polarization-shift keying (PoSK), in which information are encoded in thepolarization state e of each bit.
The PoSK modulation is not practical for optical fibers. Most digital lightwavesystems use ASK which is also known as intensity modulation. Multilevel intensitymodulation results in a significant increase in required SNR. The simplest way toapply intensity modulation is changing the signal intensity between zero and anotherlevel which is often called on-off keying (OOK). But as binary modulation formatshave a low spectral efficiency, some advanced modulations with higher spectral effi-ciency are considered recently. Using coherent detection with multilevel modulationformats increases complexity but leads to higher spectral efficiency. QPSK andDifferential QPSK (DQPSK) have attracted more attention among the advancedmodulation formats as they have reasonable complexity and a higher tolerance tofiber chromatic dispersion, polarization mode dispersion and nonlinearities [3].
M-ary quadrature amplitude modulation (QAM) is a widely used modulation formatin communication systems as it utilizes the signal space more than other modulationformats when the number of symbols increases. The optical QAM format can be
6 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
−LASER +
−
Pulse
Shaper
Mapper
Mapper
Pulse
Shaper
Input bit stream
010010011...01
I/Q Modulator
Ein Eout
d1(t)
d2(t)
an bn
π2
Figure 2.2: Block diagram of optical I/Q modulator [3].
generated with an I/Q modulator structure shown in Figure 2.2. The I/Q modu-lator can be generated by using two MZMs, each acts as an amplitude modulator.Considering Equations (2.2) and (2.3), the transfer function of the I/Q modulatorcan be written as
Eout(t) =Ac
2cos(
π
Vπ
d1(t)) cos(ωct+ ϕ)− Ac
2cos(
π
Vπ
d2(t)) sin(ωct+ ϕ) (2.4)
=Ac
2cos
[
π
Vπ
∑
n
(anp(t− nTs))
]
cos(ωct + ϕ)− Ac
2cos
[
π
Vπ
∑
n
(bnp(t− nTs))
]
sin(ωct + ϕ)
where d1(t) and d2(t) are the drive signals to the two MZMs, an and bn are themappers outputs, p(t) is the pulse-shaping filter discussed in the next section, andTs is the symbol duration.
This type of modulator is also called the Cartesian MZM. The advantage of CartesianMZM, compared with dual-drive MZM, is that linear impairments in the fiber, suchas dispersion, can be compensated by deploying only linear filtering (Chapter 3),while with the dual-drive MZM nonlinear filtering is required. Another advantageof Cartesian MZM is introducing less errors because of limited resolution of DSP [4].However they are more expensive and complex. They introduce higher optical lossand they typically need more drive power to minimize loss [5]. In this thesis we onlyapply Cartesian MZMs (i.e. I/Q modulators) and we consider different modulationschemes such as OOK, BPSK, QPSK (which is identical to 4-QAM), and 16-QAM.Using multi level modulation formats such as 16-QAM instead of conventional binarymodulations can significantly increase bandwidth efficiency.
2.1 OPTICAL TRANSMITTER 7
2.1.3 PULSE-SHAPING FILTERS
The waveform of the transmitted pulses is usually changed before sending it into thechannel. This process is done to limit the effective bandwidth of the transmission,and have a better control over the intersymbol interference caused by the channel. Sothe pulse-shaping filter determines the spectrum of the transmission. Pulse-shapingfilters need to satisfy certain criteria so that the filter itself does not introduceISI. Common criteria for evaluating filters are Nyquist ISI criterion and orthogonalpulse criterion [6, pp. 13–29]. The Nyquist pulses are appropriate for samplingreceivers whereas the orthogonal pulses are usually used as pulse-shaping filters inthe transmitter and as matched filters in the receiver. Some common pulse-shapingfilters are as follows:
• Raised-cosine (RC in the time domain) filter: The RC in time domain hasinfinitely wide bandwidth, therefore its practical applications are limited tothe communication channels that have some extra bandwidth. The RC intime domain is a Nyquist pulse and has the following equation:
p(t) =
1, |t| ≤ Ts
2(1− α)
cos2[
π42|t|−Ts(1−α)
αTs
]
, Ts
2(1− α) < |t| ≤ Ts
2(1 + α)
0, |t| > Ts
2(1 + α)
(2.5)
where Ts is the symbol duration, α is the roll-off factor, and 0 < α < 1.
• Boxcar filter: Such signals are the simplest possible waveforms as they arenaturally created by digital electronics. The boxcar filter, like the RC intime domain filter, is a Nyquist pulse and has an infinitely wide bandwidth.In optical fibers or even in twisted pair cables, where enough bandwidth isavailable, it makes sense to use such a simple pulse to reduce the limitationthat electronic equipments cause for data transmission rate. But in general,choosing a wide-band pulse-shaping filter in not a wise choice since the low-pass filter in the receiver side cuts the signal bandwidth which leads to an ISIin the system.
• Root raised-cosine (RRC) filter: This filter, together with a matched filterat the receiver side, is widely used in practical implementations. The roll-offfactor parameter, α, determines the excess bandwidth. Smaller values of αlead to less spectral efficiency and less complex filters. The RRC pulse is an
8 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
orthogonal pulse and has the following time response:
p(t) =
1√Ts
(
1− α + 4απ
)
, t = 0
α√2Ts
[(
1 + 2π
)
sin(
π4α
)
+(
1− 2π
)
cos(
π4α
)]
, t = ± Ts
4α
1√Ts
sin[π(1−α) tTs]+ 4α
Tst cos[π(1+α) t
Ts]
πTs
t[
1−(4α tTs)2] , otherwise
(2.6)
• Gaussian filter: The impulse response of a Gaussian filter is a Gaussian func-tion given by
p(t) =1
σ√2π
e−t2
2σ2 , (2.7)
where σ is the standard deviation. The Fourier transform of the Gaussian pulseis a Gaussian function. The Gaussian pulse is neither a Nyquist pulse nor anorthogonal pulse, but if we consider that a high proportion of the pulse energyis included in one symbol time, this pulse can be approximately considered asboth Nyquist and orthogonal pulses for sufficiently small σ.
• E33 pulse [7]: This kind of pulse is limited to one symbol time, Ts, thus it isboth a Nyquist and an orthogonal pulse. The transfer function is given by
p(t) =
1√E33
sin(
π2
[
1 + sin(
πtTs
)])
, −Ts
2≤ |t| ≤ Ts
2
0, otherwise
(2.8)
where E33 is the energy of the pulse.
In this thesis different pulse-shaping filters such as RRC filter, Gaussian filter, RCin the time domain filter, and E33 filter are considered. Figure 2.3 shows the time-domain representation of these filters with unit energy, and their correspondingfrequency responses are also shown in Figure 2.4.
2.2 THE OPTICAL CHANNEL
As briefly discussed previously, using fiber as the transmission medium in fiber-optic communication systems introduces different impairments such as fiber losses,dispersion, and nonlinear effects compared to wireless channels. This report onlyfocuses on dispersion in SMFs. Fiber dispersion, which is a linear phenomenon, isstudied more carefully below.
2.2 THE OPTICAL CHANNEL 9
0−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Time
Am
plitu
de
RRCGaussianRC in time domainE33
4Ts3Ts2TsTs−4Ts −3Ts −2Ts −Ts
Figure 2.3: Time response of different unit-energy pulse-shaping filters: (RRC pulsewith α = 0.75, Gaussian pulse with 80% of its energy contained in the symbol time,RC in time domain with α = 0.75, and E33 pulse).
0
0
1
2
3
4
5
6
7
Frequency
Am
plitu
de
RRCGaussianRC in time domainE33
4Ts
3Ts
2Ts
1Ts
−4Ts
−3Ts
−2Ts
−1Ts
Figure 2.4: Frequency response of different unit-energy pulse-shaping filters: (RRCpulse with α = 0.75, Gaussian pulse with 80% of its energy contained in the symboltime, RC in time domain with α = 0.75, and E33 pulse).
2.2.1 CHROMATIC DISPERSION
Optical fibers introduce two kinds of dispersion, intermodal and intramodal disper-sion, each of them leads to pulse broadening inside the fiber. Intermodal or modaldispersion happens because the signal propagation speed is not the same for all
10 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
modes of fiber. This kind of dispersion only occurs in MMFs and that is why mostfiber-optic communication systems do not use MMFs. Intramodal dispersion, whichis known as group velocity dispersion (GVD), chromatic dispersion, or simply fiberdispersion, exists in both SMFs and MMFs. There are two kinds of dispersion thatcontribute in intramodal dispersion, namely material dispersion and waveguide dis-persion. Material dispersion happens because different wavelengths that enter thefiber at the same time, travel at different velocities in the fiber and exit the fiber atdifferent times. This is because the refractive index of the fiber core is frequency-dependant. Therefore material dispersion depends on the source spectral widthand is less at longer wavelengths. The reason for waveguide dispersion is that thepropagation constant, β, is a function of fiber core size and wavelength. Moreoverthe refractive indexes of the core and the cladding are different, so light propagatesdifferently in them.
Chromatic dispersion becomes more serious at higher bit rates (> 10 Gb/s) andhighly degrades the system efficiency in the way that it limits the information ca-pacity and transmission distance in the fiber-optic channel due to pulse broadening.As mentioned before, pulse broadening stems from frequency dependence of β. Itis useful to write the Taylor-expansion of propagation constant, β(ω), around thecarrier frequency ωc:
β(ω) = β(ωc) + β ′(ωc) (ω − ωc) +1
2!β ′′(ωc)(ω − ωc)
2 +1
3!β ′′′(ωc)(ω − ωc)
3 + . . .
(2.9)
= β0 + β1 (ω − ωc) +1
2!β2(ω − ωc)
2 +1
3!β3(ω − ωc)
3 + . . .
where βm = ∂mβ
∂ωm |ω=ωc and the derivatives are taken with respect to angular frequencyω. Considering the above equation, we have the following definitions and relations [2,pp. 40–47]:
Wave number: β0 =∂β
∂ω|ω=ωc =
2π
λ(2.10)
GVD parameter: β2 =∂2β
∂ω2|ω=ωc (2.11)
Phase velocity: vp =c
n=
ω
β0
(2.12)
Group velocity: vg =
(
∂β
∂ω
)−1
(2.13)
Dispersion parameter: D =∂
∂λ
(
1
vg
)
= −2πc
λ2β2 (2.14)
Second-order dispersion parameter: S =
(
2πc
λ2
)2
β3 +
(
4πc
λ3
)
β2 (2.15)
2.3 THE OPTICAL COHERENT RECEIVER AND DETECTION 11
where λ is the operating wavelength, n is the refractive index, and c is the free-space speed of light. The dispersion parameter, D, is often known for a fiber andits typical value is 17 ps/(km.nm) for a SMF. The GVD parameter, β2, is usuallyexpressed in units of ps2/km.
2.2.2 TIME AND FREQUENCY RESPONSE
If we ignore polarization mode dispersion and nonlinearities of the fiber-optic channeland only consider the chromatic dispersion, the channel can be modeled as a linearchannel. The transfer function of a fiber of length d is given by
H(ω) = e−jβ(ω)d. (2.16)
Considering Equation (2.9), the constant term and the linear term do not causedispersion. It can also be shown that the third and higher order terms have a smallrole in dispersion compared with the second order term [8]. Thus, by replacingEq.(2.9) in Eq.(2.16) and setting ωc = 0, the baseband transfer function of CD canbe approximately written as
H(ω) = e−jβ2d2
ω2
= ejDλ2d4πc
ω2
. (2.17)
It can be seen that CD is an all-pass filter with gain one. Taking the inverse Fouriertransform of equation above gives the impulse response of CD as
h(t) =
√
1
j2πβ2dej t2
2β2d =
√
jc
Dλ2de−j πc
Dλ2dt2 . (2.18)
So the impulse response of the dispersive fiber is linear time-invariant, infinite intime, and noncausal. The real and imaginary part of this chirp are shown in Figure2.5.
The constellation diagram of the I/Q modulator (Fig. 2.2), employing differentpulse-shaping filters, with QPSK and 16-QAM modulation formats, and the effectof CD on the constellation diagram at different distances are shown in Figures 2.6–2.8. As CD greatly depends on the previous and future transitions, the effect ofCD varies by changing the transition pattern. These figures also show that the CDeffect is more severe at larger fiber lengths.
2.3 THE OPTICAL COHERENT RECEIVER ANDDETECTION
The optical receiver goal is to recover the source data that is transmitted throughthe lightwave system. To do that, first the optical signal should be converted intoan electrical signal. This task can be done via using any type of photodetectors
12 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
(a) Real part (b) Imaginary part
Figure 2.5: Real and imaginary part of chromatic dispersion in time domain.
such as photodiodes. A noncoherent receiver with direct detection only detects theintensity (amplitude) of the signal. This type of receiver is very simple as it does notneed any phase or frequency information. But as previously mentioned in Chapter1 and Section 2.1.1, binary modulation formats, such as OOK, with direct detectionhave low spectral efficiencies. Multilevel intensity modulations with direct detectionalso result in a significant increase in required SNR. So some advanced modulationschemes with higher spectral efficiencies, such as M-PSK and M-QAM, are consid-ered recently. These multilevel modulations need coherent detection. In a coherentreceiver the carrier phase is recovered by receiver circuitry and the phase informa-tion is used in demodulating the received signal. Although coherent modulationincreases the spectral efficiency and usually requires less power than noncoherentmodulation to achieve a given error rate, it also increases the complexity of thereceiver.
Figure 2.9 shows the block diagram and the equivalent mathematical model of a ho-modyne coherent I/Q demodulator which consists of an optical amplifier, an opticalbandpass filter, optical hybrids, photodiodes, electrical lowpass filters, and samplingand detection units [3].
Photodiodes perform squaring operation which causes losing the phase information.In a coherent detection system the phase information is recovered by mixing the in-coming signal with a continuous-wave reference signal from a local oscillator (LO).This is done in the optical hybrids which are key components for coherent receivers.By employing linear interconnected dividers and combiners and performing differentadditions of the reference signal, π/2 phase shifted reference signal, and the incom-ing signal, the optical hybrids deliver the inphase and quadrature signals. Opticalhybrids can be used for either homodyne or heterodyne detection. In a homodynedetection the local oscillator signal is synchronized in frequency to the carrier of theincoming signal while in a heterodyne detection there is a difference between the
2.3 THE OPTICAL COHERENT RECEIVER AND DETECTION 13
(a) d=0 km (b) d=0 km
(c) d=30 km (d) d=30 km
(e) d=60 km (f) d=90 km
Figure 2.6: Effect of CD on the QPSK (left) and 16-QAM (right) constellationdiagrams using Gaussian pulse-shaping filter (80% of the pulse energy contained ina symbol time). β2 = 22 ps2/km.
14 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
(a) d=0 km (b) d=0 km
(c) d=30 km (d) d=30 km
(e) d=60 km (f) d=60 km
Figure 2.7: Effect of CD on the QPSK (left) and 16-QAM (right) constellationdiagrams using RC pulse-shaping filter (α = 0.75). β2 = 22 ps2/km.
2.3 THE OPTICAL COHERENT RECEIVER AND DETECTION 15
(a) d=0 km (b) d=0 km
(c) d=30 km (d) d=30 km
(e) d=60 km (f) d=60 km
Figure 2.8: Effect of CD on the QPSK (left) and 16-QAM (right) constellationdiagrams using E33 pulse-shaping filter. β2 = 22 ps2/km.
16 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
Amp Optical
BPF
Optical
Hybrid
Photodiode
Photodiode
Electrical
LPF
Sampling
& Decision
+
−
LO
Optical
Hybrid
Photodiode
Photodiode
Electrical
LPF
Sampling
& Decision
+
−rLO(t)
r(t)
yI(t)
yQ(t)
ejπ2
(a) Block diagram
| |2
| |2
+
+
+
−
| |2
| |2
+
+
+
−
+
+
−
−
LOrLO(t)
r(t)
G n(t)
HBP(f)
HLP(f)
HLP(f)yI(t)
yQ(t)
YI
YQ
ejπ2
(b) Mathematical model
Figure 2.9: Coherent optical I/Q demodulator [3].
carrier frequency and the LO signal frequency.
The optical amplifier which is commonly used is the Erbium doped fiber amplifier(EDFA). The EDFA amplifier, with gain G, contributes in some optical noise, n(t),which is modeled as an additive white Gaussian noise (AWGN) with single-sidedpower spectral density N0 =
F2hvG, where h is Planck’s constant, v is the frequency
of the light, and F is the noise figure. The signal-to-noise ratio (SNR) at theamplifier output is less than the amplifier input. Noise figure shows how much noisethe amplifier adds to the input signal and is defined as F = SNRin
SNRout, in which SNRin
and SNRout are the SNR at the amplifier input and output respectively [2, Chap. 6].
A Gaussian filter is used as the optical bandpass filter for filtering out the noisecaused by EDFA. The baseband frequency response of such a filter for a passbandsystem with 3-db bandwidth B is [3]
HBP(f) = e−2loge2
B2 f2
. (2.19)
The purpose of the electrical lowpass filter, HLP, is to filter out the high frequencycomponents after the photodiode operation. There is a trade off in choosing thebandwidth of the bandpass and lowpass filters. Larger bandwidths of the filters
lead to more noise in the system and consequently the performance of the systemdegrades. On the other hand filter bandwidths lower than the signal bandwidthresult in distortion and ISI. In this thesis we use a lowpass filter which is matchedto the pulse-shaping filter.
Considering the mathematical model of the homodyne coherent receiver shown inFigure 2.9b and considering no noise in the system we can write the following ex-pressions:
r(t) = A(t) cos (ωct + φ(t)) (2.20)
rLO(t) = ALO cos (ωct) . (2.21)
where r(t) is the received signal, A(t) and φ(t) are the amplitude and phase of thereceived signal, ωc is the carrier frequency, and ALO is the amplitude of the localoscillator. Also, after lowpass filtering the photodiodes outputs we get the desiredinphase and quadrature signals given by
yI(t) = ALOA(t) cosφ(t) (2.22)
yQ(t) = ALOA(t) sinφ(t). (2.23)
Then these filtered signals are sampled at maximum eye opening and decisions aremade base on the samples.
2.4 LITERATURE STUDY ON CHROMATIC DIS-PERSION COMPENSATION
Chromatic dispersion can be compensated in either optical or electrical domain.The optical domain compensation is done via using dispersion compensating fibers(DCF) [2, pp. 436–437] or optical equalizing filters such as interferometric filters [2,pp. 438–440] or fiber Bragg gratings [2, pp. 441–448]. The shortcomings with theoptical compensation are high loss, degraded performance, costly optical compen-sators and amplifiers, and sizeable equipments. Thus compensating fiber dispersionby using electronic digital signal processing (DSP) avoids the requirement for bulky,expensive and high loss optical dispersion compensation components, as well asproviding adaptive filtering methods and advantages of high functionality, simplerdeployment and reconfiguration, and reproducibility of the system [9] [10]. High bitrates in optical communications put limitation in the use of DSPs. However, todaythe continuing rapid development of integrated circuits and digital technology al-lows implementing DSPs that can operate in such high rates with acceptable powerconsumption.
As we saw in Section 2.2.2, dispersion can be approximately considered as a linearoperation. So it can be compensated with a linear filter that mimics the inversechromatic dispersion response. Actually electronic dispersion compensation (EDC)has been accomplished with different approaches: using linear FIR filters [8, 10] at
18 Chapter 2 FIBER-OPTIC COMMUNICATION SYSTEM
transmitter or receiver side or using nonlinear filters based on look-up tables (LUTs)at the transmitter [9]. Digital infinite impulse response (IIR) filtering as a means forEDC is also proposed in [11] which claims using a significantly smaller number oftaps compared to FIR filters. Furthermore, real-time systems employing linear andnonlinear filters in a single-mode fiber have also been implemented in [12] and [13]respectively.
Performing EDC at the receiver has a limited performance in direct-detection sys-tems as the optical phase information is lost due to square operation of photodetector [14]. However some systems have been proposed based on feed-forwardequalizers (FFE) and decision-feedback equalizers (DFE) to compensate CD afterdirect-detection [15, 16]. Over the last few years, transmitter-side implementationof EDC has attracted a lot of attention since electronic predistortion (EPD) is asimpler technique and results in less complex receivers which is more desirable [14].
Large amount of required memory is a main drawback in LUT-based systems asthe memory size is exponentially proportional to the amount of dispersion, whilethe size of the linear filter scales linearly with the dispersion of the link [14]. How-ever exploiting the LUT approach to compensate nonlinear effects such as SPM isinevitable [4]. Therefore, a reasonable approach is to compensate CD by linear fil-tering and compensate nonlinear effects by LUT-based nonlinear filters [14, 17]. Asin nonlinearity compensation only a limited number of adjacent bits needs to betaken into account, the size of LUT will not be so large in this case.
In the next chapter we consider EDC using a real-time FIR filter. The filter canbe employed in either the transmitter side (precompensation) or the receiver side(postcompensation). In this method compensation is done by (pre or post) distortingthe signal using a linear filter whose transfer function is the inverse of CD, thereforeat the receiver we will have the desired signal. In Chapter 4 we discuss furtherresults on EDC using a nonlinear filter, where compensation is accomplished bypredistorting the signal via using a LUT in the transmitter side.
3 CHROMATIC DISPERSION COMPEN-SATION BY LINEAR FILTERING
As mentioned before, CD is a linear phenomenon and can be compensated by using aFIR or an IIR linear filter. The linear filter can be deployed in either the transmitterside or the receiver side; the former is named precompensation while the later isknown as postcompensation. In both pre and post compensation methods the filterapplies the inverse effect of chromatic dispersion and consequently we have thedesired signal without dispersion at the receiver.
3.1 SYSTEM MODEL
By adding a linear compensating filter to the optical transmitter, previously shownin Figure 2.2, the structure of a precompensating transmitter system can be designedas shown in Figure 3.1. The I/Q modulator transfer function is given by Equation(2.4).
Compensation is done by pre-distorting the signal using the real-time linear FIRcompensator which has the inverse chromatic dispersion transfer function. Thereforefrom Equations (2.17) and (2.18), the frequency and time response of the ICD filtercan be respectively expressed as
Hicd(ω) = ejβ2d2
ω2
(3.1)
hicd(t) =
√
j
2πβ2de−j t2
2β2d (3.2)
3.2 NUMBER OF FILTER TAPS
The design of a CD compensating filter in time domain and an upper bound onthe number of taps required to compensate chromatic dispersion is obtained in [18].
Apm
Apm
DAC
Mapper
Linear FIR
Compensator
(ICD)
Symbols
To the channelPulse
shaper
G
G
DAC
DFB +
MZMInput bit stream
010010011...01
MZM
π
2M
Vπ
πcos−1(·)
Vπ
πcos−1(·)
Figure 3.1: Schematic of the transmitter with linear FIR compensator.
20 Chapter 3 CHROMATIC DISPERSION COMPENSATION BY LINEAR FILTERING
With a similar method the minimum number of taps required to implement thecompensating filter in time domain can be derived.
As the time response in Equation (3.2) and Figure 2.5 show, the frequency of thechirp increases continuously with time. Thus, to be able to sample it, first we haveto window it. If we assume the double-sided bandwidth of the pulse shape is fp, thewindow length should be chosen in a way that at least the maximum frequency ofthe pulse shape, fp
2, is met in the chirp (see Figure 2.4), and the highest frequency
component of the chirp, fm, must be at most equal to half the sampling frequency,fs, according to Nyquist sampling criterion. So we should have
fp2
≤ fm ≤ fs2
(3.3)
Taking the derivative of the phase of Equation (3.2) with respect to time gives theangular frequency. So fm can be expressed as [18]
fm =1
2π
d
dt(
t2
2β2d)|t=tm =
tm2πβ2d
(3.4)
where tm is the time at the far end of the truncated impulse response. From theabove two equations we get
πβ2dfsfp ≤ tm∆t
≤ πβ2df2s (3.5)
where ∆t = 1fs.
If we assume the one-sided number of taps is n1, i.e. n1∆t ≤ tm ≤ (n1 + 1)∆t, thenthe total number of taps will be N = 2n1 + 1, and we have the following criterionsdirectly derived from Inequality (3.5)
⌈πβ2dfsfp⌉ ≤n1 ≤⌊
πβ2df2s
⌋
(3.6)
2× ⌈πβ2dfsfp⌉ + 1 ≤N ≤ 2×⌊
πβ2df2s
⌋
+ 1 (3.7)
where ⌈x⌉ is the nearest integer greater than or equal to x and ⌊x⌋ is the integerpart of x.
According to the latest inequality the bounds for the number of filter taps arefunctions of sampling frequency. By considering the minimum sampling frequencyallowed, i.e. fs = fp, the minimum possible number of taps that can be used toimplement the CD compensating filter can be expressed as follows
N = 2×⌊
πβ2df2p
⌋
+ 1 (3.8)
Thus the minimum possible number of taps is a function of pulse shape bandwidth,fp. The bandwidths of some common pulse shapes are compared in Figure 2.4. Theminimum possible number of taps given by above equation are also illustrated inFigure 3.2 using different pulse shapes and at different fiber lengths. The bandwidthof the RRC pulse is 1+α
Ts. The bandwidth of the Gaussian pulse (with 80% of its
3.3 TIME-DOMAIN AND FREQUENCY-DOMAIN IMPLEMENTATIONS 21
500 1000 1500 2000 2500 3000 3500 4000 4500 50000
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
Fiber length (Km)
Min
imum
num
ber
of r
equi
red
filte
r ta
ps
RRCGaussianRCE33
Figure 3.2: The minimum number of taps required at different fiber lengths for a FIRcompensating filter in a 10 Gbaud/s optical transmitter. β2 = 22 ps2/km. The pulse-shaping filters used are: (RRC pulse with α = 0.75 and fp = (1 + α)/Ts, Gaussianpulse with 80% of its energy contained in the symbol time and fp = 2.44/Ts, RC intime domain with α = 0.75 and fp = 6/Ts, and E33 pulse with fp = 5/Ts).
energy contained in one symbol time) is chosen to be 2.44Ts
so that it contains 99% of itsenergy. The bandwidths of RC (in time domain) and E33 pulse are also considered6Ts
and 5Ts
respectively. This figure shows that using a narrow-band pulse such asRRC, can decrease the minimum possible number of taps and thus the complexityof the filter, for example for over 5000 km of fiber with β2 = 22 ps2/km in a 10Gbaud/s system only around 200 taps are needed if the RRC pulse is used.
Truncating the impulse response to less than the limit given by Equation (3.7)destroys the signal bandwidth and leads to distortion and ISI. On the other handgoing further the upper limit causes aliasing as the Nyquist sampling theorem is notmet anymore.
3.3 TIME-DOMAIN AND FREQUENCY-DOMAINIMPLEMENTATIONS
As mentioned before, the linear compensating filter can be implemented in eithertime or frequency domain. The time-domain implementation and the required num-ber of taps are discussed in the previous section. For the frequency-domain imple-mentation the Equation (3.1) is directly employed, sampled and multiplied in the
22 Chapter 3 CHROMATIC DISPERSION COMPENSATION BY LINEAR FILTERING
−1.5
−1
−0.5
0
0.5
1
1.5
(a) d = 50 km
−1.5
−1
−0.5
0
0.5
1
1.5
(b) d = 5000 km
Figure 3.3: The effect of windowing the compensating filter on the eye diagram ofa QPSK system at a lower fiber length (left) compared to the negligible windowingeffect at a higher fiber length (right). Red points are the symbol times.
transfer function of the filter input. In contrast to the time-domain implementationof the compensating filter, there is no upper limit for the number of filter taps in fre-quency domain since the input signal to the compensating filter is band-limited. Tohave a fair comparison between time and frequency domain implementations of thecompensating filter (i.e. to have the same number of filter taps for both), the linearcompensating filter is implemented for realtime processing by using the overlap-savetechnique. In this technique the large input stream to the filter is broken into blocksof shorter lengths. The outputs of successive blocks are computed and concatenatedin a proper way to generate the total output. This method also reduces the runtimesignificantly.
Windowing effect in time-domain implementation of the filter is inevitable whenthe number of taps is not large enough and this causes a small deviation from theactual transfer function of ICD filter given by (3.1). Figure 3.3 shows the windowingeffect of implementing the compensating filter in time domain on the eye diagramwhen the minimum number of taps given by Equation (3.7) is used in a QPSKsystem. Figure 3.4 also shows the eye diagrams at the same fiber lengths, but herethe compensating filter is implemented in the frequency domain. It can be seen inFigure 3.3a that at lower fiber lengths the eye is more closed compared to the eyeat higher lengths when the time-domain filter is used (Figure 3.3b), or compared tothe eye diagram when the frequency-domain filter at the same filter length is used(Figure 3.4a). Since at higher lengths of fiber the number of taps is large and thetruncation length is long enough, the windowing effect is negligible.
3.4 EFFECT OF FILTERING ON THE AMPLITUDE 23
−1.5
−1
−0.5
0
0.5
1
1.5
(a) d = 50 km
−1.5
−1
−0.5
0
0.5
1
1.5
(b) d = 5000 km
Figure 3.4: Eye diagram of a QPSK system at two different fiber lengths when thecompensating filter is implemented in frequency domain. Red points are the symboltimes.
3.4 EFFECT OF FILTERING ON THE AMPLITUDE
The cosine function of I/Q modulator, as stated in equation (2.4), is compensatedby using an inverse cosine block before it, as shown in Figure 3.1. This can beimplemented using a small LUT. As the input to the inverse cosine function can onlyhave the values between -1 and 1, the amplitude of the signal after the compensatingfilter must be normalized to fall within this range which is done by block G inFigure 3.1. Here for normalizing the amplitude, we need to calculate the maximumamplitude generated after the compensating filter. The output of the filter can bewritten as
r =L−1∑
n=0
sn (p(t− nTs) ∗ hicd(t)) =L−1∑
n=0
snx(t− nTs) (3.9)
where sn is n’th transmitted symbol, L is the total number of symbols to be transmit-ted, Ts is the symbol duration, p(t) is the pulse shape, x(t) = p(t) ∗ hicd(t), and hicd(t)is the inverse time response of CD given by (3.2). By assuming sn = sn,r + jsn,i andx(t) = xr(t) + jxi(t) the maximum amplitude of the inphase signal can be expressedas
Amax,inphase = max(Re{r}) =max
(
L−1∑
n=0
sn,rxr(t−nTs)−sn,ixi(t−nTs)
)
(3.10)
The maximum amplitude of the quadrature signal can be derived in a similar way.By symmetry we can say the maximum amplitude of the inphase signal is the same
24 Chapter 3 CHROMATIC DISPERSION COMPENSATION BY LINEAR FILTERING
as the quadrature signal and is given by
Amax,inphase = Amax,quadrature = max1≤l0≤ls
{ K∑
n=1
B[
∣
∣xr(l0 + (n− 1)ls)|+ |xi(l0 + (n− 1)ls)∣
∣
]
}
(3.11)
where B = max (sn,r) = max (sn,i) and depends on the modulation scheme. ForQPSK and 16-QAM modulations B is typically 1 and 3 respectively. ls =
⌈
Ts
∆t
⌉
which is the number of samples per symbol and
K =
⌈
N∆t + Tp
Ts
⌉
(3.12)
in which ⌈x⌉ is the nearest integer greater than or equal to x, Tp is the time durationof pulse p(t) and N is defined according to equation (3.7). K can also be interpretedas the number of adjacent symbols that contribute to the ISI.
A specific input bit pattern results in the maximum amplitude derived by (3.11).The probability that such a pattern occurs can be very low as this probabilitydecreases exponentially by increasing dispersion depth (K). This probability canbe expressed as PAmax = 2
MK where M is the number of constellation points (e.g.,M = 4 in a QPSK and M = 16 in a 16-QAM system) and K, as calculated above,depends on time duration of the compensating filter and the pulse p(t).
The effect of compensating filter on the maximum amplitude of the signal is illus-trated in Figure 3.5. Figures 3.5a and 3.5b show the ratio Amax
Apfor 16-QAM, QPSK,
BPSK, and OOK modulation schemes at different fiber lengths, where Ap and Amax
are the maximum amplitudes of the signal before and after filtering respectively.The simulations are done with different pulses: The RRC pulse with α = 0.75, theGaussian pulse with 80% of its energy included in each symbol duration, RC (in timedomain) with α = 0.75, and the E33 pulse. Figures 3.5c and 3.5d also show themaximum amplitude generated after the compensating filter, if unit energy pulsesare used.
3.5 PERFORMANCE OF LINEAR PRECOMPEN-SATING METHOD
The simulated performance of the QPSK and 16-QAM systems at different fiberlengths using the system model of Figure 3.1 are shown in Figures 3.6 and 3.7 re-spectively. As the curves show, the performances of each system is the same as whattheory indicates (regardless of filter implementation in time or frequency domain),which means that the fiber dispersion is completely compensated. The theory curvesindicate the performance of the systems using Gray coding and operating in whitenoise when there is no dispersion or in other words when the the fiber length is setto zero.
3.5 PERFORMANCE OF LINEAR PRECOMPENSATING METHOD 25
0 500 1000 1500 2000 2500 3000 3500 4000 45000
2
4
6
8
10
12
14
Fiber length (km)
Am
ax/A
p
RRC, QPSKGaussian, QPSKRC, QPSKE33, QPSKRRC, 16QAMGaussian, 16QAMRC, 16QAME33, 16QAM
(a)
0 500 1000 1500 2000 2500 3000 3500 4000 45000
2
4
6
8
10
12
14
Fiber length (km)
Am
ax/A
p
RRC, BPSKGaussian, BPSKRC, BPSKE33, BPSKRRC, OOKGaussian, OOKRC, OOKE33, OOK
(b)
0 500 1000 1500 2000 2500 3000 3500 4000 45000
1
2
3
4
5
6
Fiber length (km)
Am
ax
RRC, QPSKGaussian, QPSKRC, QPSKE33, QPSKRRC, 16QAMGaussian, 16QAMRC, 16QAME33, 16QAM
(c)
0 500 1000 1500 2000 2500 3000 3500 4000 45000
1
2
3
4
5
6
Fiber length (km)
Am
ax
RRC, BPSKGaussian, BPSKRC, BPSKE33, BPSKRRC, OOKGaussian, OOKRC, OOKE33, OOK
(d)
Figure 3.5: The effect of compensating filter on amplitude of the signal in differentmodulation schemes using RRC (α = 0.75), Gaussian (80% of its energy includedin each symbol duration), RC in time domain (α = 0.75), and E33 pulses.
3.5.1 THE EFFECT OF D/A RESOLUTION ON THE PER-
FORMANCE
According to the system model in Figure 3.1, a digital-to-analog converter (DAC) isused after the inverse cosine block to generate the analog drive signals of the MZMsfrom digital samples of the predistorted signal. The analog value of the signal afterthe DAC will always differ from the quantized digital value before and the differenceis called quantization error. Higher resolutions (i.e. the number of bits, b, used foreach sample) in DACs leads to less quantization error. Figures 3.8 and 3.9 showthe effect of DAC resolution on the performance of the QPSK and 16-QAM linearprecompensation systems at 5000km fiber length respectively. It can be seen that
26 Chapter 3 CHROMATIC DISPERSION COMPENSATION BY LINEAR FILTERING
1 2 3 4 5 6 7 8 9−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Eb/N0 (dB)
log1
0BE
R
Theory500 km5000 km
Figure 3.6: BER vs. SNR in a QPSK linear precompensation system at 0km, 500km,and 5000km fiber lengths.
1 2 3 4 5 6 7 8 9 10 11 12 13−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
Eb/N0 (dB)
log1
0BE
R
Theory500 km5000 km
Figure 3.7: BER vs. SNR in a 16-QAM linear precompensation system at 0km,500km, and 5000km fiber lengths
in the QPSK system using a 4-bit or higher resolution DAC does not degrade theperformance. For the 16-QAM system 1 more bit is needed as the input range to
3.5 PERFORMANCE OF LINEAR PRECOMPENSATING METHOD 27
1 2 3 4 5 6 7 8 9−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Eb/N0 (dB)
log1
0BE
R
Infinite resolution2−bit DAC3−bit DAC4−bit DAC5−bit DAC
Figure 3.8: BER comparison of a QPSK system using 2, 3, 4, and 5-bit DAC at5000km fiber lengths.
1 2 3 4 5 6 7 8 9 10 11 12 13−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
Eb/N0 (dB)
log1
0BE
R
Infinite resolution2−bit DAC3−bit DAC4−bit DAC5−bit DAC
Figure 3.9: BER comparison of a 16-QAM system using 2, 3, 4, and 5-bit DAC at5000km fiber length
the DAC is larger compared to the QPSK system. Here, with a 5-bit DAC we don’tlose the performance.
4 CHROMATIC DISPERSION COMPEN-SATION BY NONLINEAR FILTERING
Look-up tables can be used as general nonlinear filters if the input signal can onlyassume a finite number of values. As mentioned before, chromatic dispersion hasalso been compensated using a look-up table [9]. The drawback with this approachis the large amount of memory required for the LUT since in CD compensation thenumber of bits that need to be taken into account can be large [14]. But on theother hand the advantage of using LUTs is that they can be used to compensatenonlinear effects such as higher order dispersions and SPM.
In [19] it is shown that the LUT generated for compensating CD in dispersion-limited links can be significantly compressed by applying the Hadamard transform.In this chapter we discuss further results on what has been done in [1] and [19].
4.1 THE TRANSMITTER SYSTEM MODEL
The system model of the transmitter with compensation employing LUT is shown inFigure 4.1. As described in [1], if we assume ls samples per symbol, then for each κinput symbols the LUT gives 2 · ls output samples corresponding to the predistortedmiddle symbol of the input sequence (the factor 2 is because of quadrature andinphase channels). Therefore by considering all input symbol patterns, we will havea LUT of size 2 · Mκ × ls where M is the number of different symbols or in otherwords the number of constellation points.
The system design for generating the LUT is shown in Figure 4.2. The inversechannel filter is the same as the compensating filter used in Chapter 3 and mimicsthe inverse behavior of the actual optical channel. The inverse cosine block is forcompensating the cosine function of the Cartesian Mach-Zehnder modulator. Asmentioned in Section 3.4, since the input to the inverse cosine can only be between-1 and 1, we should make sure that the amplitude of the signal after the inversechannel filter is always within this range.
4.2 THE CONSTRAINTS AND THE SOLUTIONS
By increasing the fiber length, the amount of dispersion increases. As a result, if weassume the baud rate is constant, more adjacent symbols interfere with each otheraccording to Equation (3.12). Hence, to generate the LUT a longer input symbolsequence should be considered. As the size of LUT is exponentially dependent onthe number of input symbols, κ, for high amounts of dispersion generating LUT
4.2 THE CONSTRAINTS AND THE SOLUTIONS 29
−LASER +
−
Input bit stream
010010011...01
I/Q Modulator
DACDAC
LUT
(I)
LUT
(Q)
Ein Eout
d1(t) d2(t)
π2
Figure 4.1: Schematic of optical transmitter with LUT-based compensator [1].
may not be feasible due to large memory requirements.
A method for compressing the LUT with the Hadamard transform has been proposedin [1] and [19] which greatly reduces the size of memory and scales it linearly withthe number of input symbols. So instead of using LUT in the system model ofFigure 4.1, the compressed look-up table (CLUT) can be replaced. More detailson how the CLUT is generated and how the reconstruction of LUT from CLUT isdone can be found on [1]. Although the main solution to the memory constraintis given by the Hadamard transform technique, there are still some constraints inimplementing such a system. Here I introduce these constraints and I discuss aboutsome implementation details to avoid some of these constraints.
−LASER +
−
Pulse
Shaper
Mapper
Mapper
Pulse
Shaper
Input bit stream
010010011...01
I/Q Modulator
Inverse CD
d1(t)
d2(t)
π2
Vπ
πcos−1(·)
Vπ
πcos−1(·)
Figure 4.2: Schematic of the system for generating LUT [1].
30Chapter 4 CHROMATIC DISPERSION COMPENSATION BY NONLINEAR FILTERING
As mentioned in [1], to generate the LUT with Mκ rows, a very long symbol se-quence, which guarantees that all the κ symbol patterns are contained in it, isgenerated. Using a pseudo random binary sequence (PRBS), in best case a bit se-quence of length logM ×Mκ is needed to include all the symbol patterns. If κ getslarge this can be a very large sequence, especially after upsampling, and it can intro-duce memory constraints for saving such a long vector. Another constraint can bethe processing time for filtering such large number of samples. Here the processingtime can be reduced by real-time processing implementation of the filter as partiallydescribed in Section 3.3.
As depicted in Figure 4.3a, after generating the long waveform corresponding tothe large input symbol sequence, the CLUT can be generated in two ways. Oneway is generating LUT first, and then applying the Hadamard transform to it toget a table of the same size but with lots of zero-value rows. Next, the nonzerorows of the transformed LUT form the compressed table, CLUT. Here, again weencounter memory constraint in generating CLUT, since the large-size LUT have tobe generated first.
To avoid the large memory requirement introduced by the LUT, we should considergenerating CLUT directly from the long predistorted waveform. As mentioned be-fore, LUT is generated in the following way that each time we look for a part of thelong predistorted waveform that corresponds to the middle symbol in the κ inputsymbol sequence and we save that part in a row of LUT defined by the κ input sym-bol sequence. After filling all the LUT rows in this way, we apply the Hadamardtransform to it to compress it. After taking the Hadamard transform of the LUT,most of the rows have zero values and they can be skipped.
Now, to avoid generating LUT we can do the following. We know what the significantHadamard components are (the nonzero rows) [1], and we know that each elementof the Hadamard matrix is given by
(H)i,j = (−1)∑
k ikjk (4.1)
where (H)i,j is the element (i, j) of the Hadamard matrix H, and ik and jk are k’thbits in the base-2 representation of i and j respectively. Each time that we find thepredistorted waveform corresponding to the middle symbol in the κ input symbolsequence, we can perform some part of required calculations for generating CLUTinstead of saving it in a row of the LUT. So for each extracted waveform from thelong predistorted waveform, some part of each CLUT element is calculated andadded to the previous value of that element. This is shown in Figure 4.3a whereCLUT is generated from the long waveform in one step.
So far we have removed the need for generating the LUT before generating CLUT.But as discussed before, still the long input symbol sequence introduces a memorysize constraint. Figure 4.3b shows another way of generating LUT. Instead of gen-erating a long bit sequence that contains all κ symbol sequence patterns, we cangenerate Mκ different sequences of length κ symbols and construct each LUT rowby applying each of these sequences in a loop.
4.2 THE CONSTRAINTS AND THE SOLUTIONS 31
LUT
010010000001101111011100101010010011000110111.........01
Memory constraint
HT−LUT
Processing−time constraint
CLUT
1
32
1
Memory constraint
Processing−time constraint
Processing−time constraint
At least (logM ·Mκ) bits
(a)
LUT
Memory constraint
HT−LUT
Processing−time constraint
CLUT
1
32
1
Memory constraint
Processing−time constraint
Processing−time constraint
..................0111...11 1011...00 0001...100101...01 Blocks of (logM · κ) bits
(b)
Figure 4.3: Generating CLUT from a long input bit stream (a) or from smaller sizeblocks of bits (b).
In a case that the number of symbols, κ, in the input symbol sequence of LUT isless than the actual number of symbols that should be considered, K in Equation(3.12), some random symbols can be concatenated to the input symbol sequence inorder not to lose the advantage of averaging that the Hadamard transform performsin generating the CLUT.
In this way of generating the LUT, again we can apply the previously discussedimplementation technique to generate CLUT directly without first generating LUT.Therefore, finally by breaking the long input sequence to smaller sequences and byrealtime calculation of CLUT elements we do not face any memory limitation in theimplementation anymore. Although still the processing time can be very large if κgets large, we should notice that this time is only spent once at the start-up.
32Chapter 4 CHROMATIC DISPERSION COMPENSATION BY NONLINEAR FILTERING
1 2 3 4 5 6 7 8 9−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Eb/N0 (dB)
log1
0BE
R
TheoryCLUTLUT
Figure 4.4: Comparing BER of QPSK systems with compression (CLUT) and with-out compression (LUT) when the block size (=14 bits) is adequate regarding thefiber length (d = 200 km).
4.3 PERFORMANCE OF NONLINEAR PRECOM-PENSATING METHOD
The LUT-based precompensation system with QPSK modulation format workingat 10 Gbaud is simulated. The input symbol sequence to the LUT is set to 7symbols, i.e. 14 bits, and CLUT is directly generated as described in the previoussection. Thus by taking 32 samples per symbol, i.e. ls = 32, the dimensions ofLUT and CLUT are 2 · 214 × 32 and 2 · (14 + 1)× 32 respectively. The performanceof the system is tested in two different scenarios. First a lower value is chosenfor the fiber length (d = 100 km) so that the symbol block size is adequate andcontains all the adjacent interfering symbols. In this case the chromatic dispersionis completely compensated if either LUT or CLUT is used. This result is shownin Figure 4.4. Then by increasing the fiber length to (d = 500 km), the systemis simulated for a case that the symbol block size is less than the actual numberof adjacent symbols that interfere with each other. As shown in Figure 4.5, inthis case there is some degradation in the performance of the system as the CDis not compensated completely. The more affected symbols by dispersion ignored,the more degradation in performance we will have. As discussed in [19] and [1], theperformance of the system using CLUT is better than the system with LUT becauseof the averaging property of the Hadamard transform.
As previously mentioned, by removing the memory constraint, larger lengths of
4.4 COMPARING WITH LINEAR COMPENSATION, PROS AND CONS 33
1 2 3 4 5 6 7 8 9−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Eb/N0 (dB)
log1
0BE
R
TheoryCLUTLUT
Figure 4.5: Comparing BER of QPSK systems with compression (CLUT) and with-out compression (LUT) when the block size (=14 bits) is inadequate regarding thefiber length (d = 500 km).
input sequence to the LUT can be applied. In Figure 4.6 we see that by increasingthe block size to 22 bits, the CD that could not be compensated in the previouslywith a lower block size, is now completely compensated.
4.4 COMPARINGWITH LINEAR COMPENSATION,PROS AND CONS
If we assume that the number of adjacent symbols affected by dispersion is K ac-cording to Equation (3.12), the total memory requirement for the nonlinear filteringmethod without compression is given by 2 ·MK lsb, where M is the number of con-stellation points, ls is the number of samples per symbol, b is the D/A resolution,and the factor 2 implies real and imaginary parts. The computational complexitycan be high at the start-up for generating the LUT if the table size is large, but theruntime complexity is very low as it only includes reading from a memory. Generat-ing a LUT for high amount of dispersion may not be feasible because of limitationsin storage capacity and also huge processing time.
The nonlinear filtering with direct compression requires much lower memory. Thetotal memory requirement is given by 2 · logM ·Klsb. The startup and runtime com-putational complexities are higher compared to the method without compression asthe Hadamard and inverse Hadamard transforms are involved in them respectively.
34Chapter 4 CHROMATIC DISPERSION COMPENSATION BY NONLINEAR FILTERING
1 2 3 4 5 6 7 8 9−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Eb/N0 (dB)
log1
0BE
R
TheoryCLUT (block size = 22)CLUT (block size = 14)
Figure 4.6: BER of QPSK systems for a larger block size (=22 bits) compared tothe previously used smaller block size (=14 bits) in 500 km fiber length.
Although more dispersion can be compensated with this approach, generating CLUTcan also be infeasible at high amounts of dispersion due to the huge computationalcomplexity.
Finally, the memory requirements for the linear filtering method is 2 ·Nb or equiva-lently 2 ·Klsb where N is the number of filter taps and is defined in Equation (3.7).The runtime complexity is higher than the nonlinear filtering method as filtering thesignal is continuously done in the transmitter. The advantage with this approach isthat even high amounts of dispersion can be completely compensated as the numberof filter taps is very high in most of the cases.
5 CONCLUSIONS
Chromatic dispersion is a linear phenomenon and can be fully compensated by usinglinear FIR filters to predistort the signal. Reasonable number of taps is needed evenfor high amounts of dispersion if a narrow-band signal is used. In the lookup table-base compensation method, chromatic dispersion can now be compensated for largerdistances as the memory constraint is removed. Although look-up tables are usuallyused to compensate the nonlinear effects of the channel in a fiber optic system,chromatic dispersion compensation by using a compressed lookup table can also beof interest if the processing time for generating CLUT is not too much because thismethod has a low runtime computational complexity compared to linear filteringmethod. For example for a system working at 10 Gbaud and at 500 km fiber lengththe CLUT can be generated in reasonable amount of time so it should be preferred tothe linear filtering method as it offers less runtime complexity. We also noticed thatCLUT improves the performance of the system compared to LUT only if inadequateblock size is chosen. Finally we saw that a 4-bit DAC in the QPSK and a 5-bit DACin the 16-QAM system do not degrade the performance.
35
Bibliography
[1] F. A. Khan, Electronic dispersion compensation by signal predistortion using acompressed lookup table. Master thesis, Department of Signals and Systems,Chalmers University of Technology, Goteborg, Sweden, 2009.
[2] G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. John Wiley andSons Inc., New York, USA, 1997.
[3] H. Zhao, On Interference Suppression Techniques for Communication Systems.PhD thesis, Department of Signals and Systems, Chalmers University of Tech-nology, Goteborg, Sweden, 2008.
[4] P. Watts, R. Waegemans, M. Glick, P. Bayvel, and R. Killey, “An FPGA-basedoptical transmitter design using real-time DSP for advanced signal formats andelectronic predistortion,” Journal of Lightwave Technology, vol. 25, no. 10, pp.3089 –3099, Oct. 2007.
[5] D. McGhan, M. O’Sullivan, M. Sotoodeh, A. Savchenko, C. Bontu, M. Be-langer, and K. Roberts, “Electronic dispersion compensation,” in Optical FiberCommunication Conference, 2006.
[6] J. B. Anderson, Digital Transmission Engineering, 2nd ed. John Wiley andSons Inc., USA, 2004.
[7] E. Ip and J. Kahn, “Power spectra of return-to-zero optical signals,” Journalof Lightwave Technology, vol. 24, no. 3, pp. 1610 –1618, March 2006.
[8] M. Said, J. Sitch, and M. Elmasry, “An electrically pre-equalized 10-Gb/sduobinary transmission system,” Journal of Lightwave Technology, vol. 23,no. 1, pp. 388 – 400, Jan. 2005.
[9] R. Killey, P. Watts, V. Mikhailov, M. Glick, and P. Bayvel, “Electronic disper-sion compensation by signal predistortion using digital processing and a dual-drive mach-zehnder modulator,” IEEE Photonics Technology Letters, vol. 17,no. 3, pp. 714 –716, March 2005.
[10] J. McNicol, M. O’Sullivan, K. Roberts, A. Comeau, D. McGhan, andL. Strawczynski, “Electrical domain compensation of optical dispersion,” inOptical Fiber Communication Conference, vol. 4, March 2005.
[11] G. Goldfarb and G. Li, “Chromatic dispersion compensation using digital IIRfiltering with coherent detection,” IEEE Photonics Technology Letters, vol. 19,no. 13, pp. 969 –971, July 2007.
[12] P. Watts, R. Waegemans, Y. Benlachtar, V. Mikhailov, P. Bayvel, and R. I.Killey, “10.7 Gb/s transmission over 1200 km of standard single-mode fiber by
36
BIBLIOGRAPHY 37
electronic predistortion using FPGA-based real-time digital signal processing,”Opt. Express, vol. 16, no. 16, pp. 12 171–12 180, Aug. 2008.
[13] R. Waegemans, S. Herbst, L. Holbein, P. Watts, P. Bayvel, C. Furst, and R. I.Killey, “10.7 Gb/s electronic predistortion transmitter using commercial FP-GAs and D/A converters implementing real-time DSP for chromatic dispersionand SPM compensation,” Opt. Express, vol. 17, pp. 8630–8640, May 2009.
[14] R. Killey, P. Watts, M. Glick, and P. Bayvel, “Electronic dispersion compen-sation by signal predistortion,” in Optical Fiber Communication Conference,March 2006.
[15] P. Watts, V. Mikhailov, S. Savory, P. Bayvel, M. Glick, M. Lobel, B. Chris-tensen, P. Kirkpatrick, S. Shang, and R. Killey, “Performance of single-modefiber links using electronic feed-forward and decision feedback equalizers,” IEEEPhotonics Technology Letters, vol. 17, no. 10, pp. 2206 – 2208, Oct. 2005.
[16] J. Wang and J. Kahn, “Performance of electrical equalizers in optically ampli-fied OOK and DPSK systems,” IEEE Photonics Technology Letters, vol. 16,no. 5, pp. 1397 –1399, May 2004.
[17] D. McGhan, C. Laperle, A. Savchenko, C. Li, G. Mak, and M. O’Sullivan,“5120-km RZ-DPSK transmission over G.652 fiber at 10 Gb/s without opticaldispersion compensation,” IEEE Photonics Technology Letters, vol. 18, no. 2,pp. 400 –402, 2006.
[18] S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express, vol. 16,pp. 804–817, Jan. 2008.
[19] F. A. Khan, E. Agrell, and M. Karlsson, “Electronic dispersion compensation byHadamard transformation,” in Optical Fiber Communication (OFC), collocatedNational Fiber Optic Engineers Conference, 2010.
